Scanning Microscopy
Volume 1990 Number 4 Fundamental Electron and Ion Beam Interactions with Solids for Microscopy, Article 5 Microanalysis, and Microlithography
1990
Plasmons in Scanning Transmission Electron Microscopy Electron Spectra
R. H. Ritchie University of Tennessee
A. Howie Cavendish Laboratory
P. M. Echenique Universidad Pais Vasco
G. J. Basbas Physical Review Letters
T. L. Ferrell University of Tennessee
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Recommended Citation Ritchie, R. H.; Howie, A.; Echenique, P. M.; Basbas, G. J.; Ferrell, T. L.; and Ashley, J. C. (1990) "Plasmons in Scanning Transmission Electron Microscopy Electron Spectra," Scanning Microscopy: Vol. 1990 : No. 4 , Article 5. Available at: https://digitalcommons.usu.edu/microscopy/vol1990/iss4/5
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Authors R. H. Ritchie, A. Howie, P. M. Echenique, G. J. Basbas, T. L. Ferrell, and J. C. Ashley
This article is available in Scanning Microscopy: https://digitalcommons.usu.edu/microscopy/vol1990/iss4/5 Scanning Microscopy Supplement 4, 1990 (Pages 45-56) 0892-953X/90$3.00+ .00 Scanning Microscopy International, Chicago (AMF O'Hare), IL 60666 USA
PLASMONS IN SCANNING TRANSMISSION ELECTRON MICROSCOPY ELECTRON SPECTRA
R.H. llitchie,• A. Howie, 1 P. M. Echenique, 2 G. J. Basbas,3 T. L. Ferrell, and J. C. Ashley
Oak llidge National Laboratory, P. 0. Box 2008, Oak Ridge, TN 37831-{3123 and Department of Physics, University of Tennessee, Knoxville, TN 37996 USA !Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, UK 2Universidad Pais Vasco, Facultad de Quimica, Aptdo 1072, San Sebastian, SPAIN 3Physical Review Letters, Box 1000, Ridge, New York 11961 USA
Abstract Introduction
A general self-energy formulation of the Surface and bulk plasmon excitations in condensed interaction between an electron in a scanning matter by swift charged particles have been much studied transmission electron microscope (STEM) and a localized for the last four decades [1-{3]. Here we describe the target is given. We prove a theorem relating the excitation of such collective modes by swift electrons that probability of energy transfer to that calculated may be formed into a beam incident on a localized target. classically. Local dielectric theory of target excitation for Using the finely focused probe in the scanning various geometries is discussed. The problem of transmission microscope (STEM), data can be collected localization of initially unlocalized excitations in the with a spatial resolution of 2 nm or better in the region of valence band of solids is treated by transforming cross surfaces, interfaces, small particles, and other features of sections differential in momentum transfer into inhomogeneous specimens. Energy analysis of the dependence on an impact parameter variable. We are inelastically scattered electrons gives much useful thereby able to account for experimental data in scanning information about the target. Such interactions involve electron microscopy (SEM) that show high spatial the wave-particle duality of the electrons in an resolution. interesting context. We introduce a generalized self-energy formulation of the electron-target interaction that describes the full quanta! properties of the probe. A general theorem relating the energy losses by an electron microprobe to those experienced by classical electrons with the same energy is described [7]. Recent progress in analyzing STEM data on energy losses in inhomogeneous targets using classical theory is reviewed. We also treat some aspects of secondary electron emission (SE) from such targets, emphasizing the localization of initially unlocalized excitations and the spatial distribution of collective modes created in the valence band of a solid.
The Self-Ener~y in STEM
It is convenient to use the self-energy concept in describing the interaction of a fast electron with an inhomogeneous target [8-13]. The essential quanta! Key Words: Plasmons, scanning transmission electron properties of the incident electron are properly treated, microscopy, inelastic interactions, electron energy losses, while the target is represented in terms of its response quanta! self-energy, localization, spatial resolution, dielectric function. Information about the spatial dependence of theory, scanning electron microscopy the interaction probability may be inferred readily from the self-energy function. A general treatment of inelastic and elastic interactions in STEM has been given [14,15]. Here we use a different emphasis couched a priori in a mixed space-energy representation. This treatment is particularly appropriate to those interactions arising in * Address for Correspondence: STEM, where information about energy transfers to localized regions of space is of interest. Here we neglect R. H. Ritchie relativistic effects and consider excitations of a target Oak Ridge National Laboratory with characteristic energies >>kT, where T is the Post Office Box 2008 temperature of the target. Oak Ridge, TN 37831-{3123 Consider the Green function G E(r,r') describing Telephone Number: (615) 574-{3208 the propagation of an electron with energy E from the
45 R.H. Ritchie, et al.
Figure l (a) Figure 2 (b) (c)
Figure 1. The Feynman diagram corresponding to the Figure 2. Feynman diagrams representing higher-order first-order self-energy of an electron interacting with a contributions to the self-energy of an electron. Figure 2a target as expressed in Eq. 2 corresponds to Eq. 5. space point r to the point r'. Many-body perturbation ,i,•(r') ,i,(r') theory [16,17] may be invoked to write a Dyson integral v(r) = e2 d3r' ----- , (4) equation for GE(r,r') in terms of Gi(r,r'), the J I r - r' I noninteracting Green function; for Coulomb interactions between electrons in the target. GE(r,r') = G~(r,r') We neglect elastic scattering for the present purposes. 3 3 The Feynman diagram of Fig. 1 illustrates the + f d rJ d r 2 GE(r,r 2) ~\(r 2,r1) Gi(r 1,r'). (1) interaction corresponding to Eq. 2. A term of next higher order in the electron-target interaction may be written, The spatially-dependent, proper self-energy, EE(r,r' ), of the propagating electron due to interaction with the target, in turn may be expressed as an infinite series, the first term of which may be written
(5)
The exact linear response function, or polarization propagator, for disturbances in the many-particle target corresponding to the Feynman diagram of Fig 2a. is given by Figures 2b and 2c illustrate two other contributions to EEof this same order. Note that the self-energy diagram "{ (Olv(r)I n) (n I v(r')IO) of Fig. 2c is obtained if we replace G by GO in Eqs. 2 and W E(r,r') = kJ 5. It does not correspond to the "proper" self-energy, n ~n + E + ia found algebraically in a straightforward manner for translationally invariant systems, and discussed in the literature (see Refs. 10,11). Generalization to still higher-order terms is straightforward [11,12]. (Olv(r')ln)(nlv(r)ID)} Analytical solution of the Dyson equation (Eq. 1 +------) (3) above) is not possible for systems without translational ~n - E + ia invariance. Here we approximate the self-energy of a projectile interacting with an arbitrary target by a series of terms that are ordered according to the number of where a is a positive infinitesimal, ( I 0), In)) is the exact times the projectile interacts with the target. In doing so state vector of the many-particle target in the (ground, we use the linear response function of the target since we nth excited) state and 6'n = 6' - in, where ( ~' 6'n) is expect that this procedure will yield reasonable results in 0 0 many cases. the energy of the (ground, nth excited) state. The The electron Green function may be approximated interaction energy v(r) may be expressed in terms of in the standard form (,i,•(r), ,i,(r)), the wave (creation, annihilation) operator tor a particle at position r in the target, viz., (6)
46 Plasmons in STEM Electron Spectra where uk(r) is an exact eigenfunction for the fully SOURCE interacting electron, Ek is the eigenenergy of this state and ois a positive infinitesimal. If the electron is prepared in a state specified by the state function 1f;(r), then the "averaged" self-energy 0 ELECTRON LENS
is complex . Its real part yields the shift in the electron's energy while the imaginary part, when multiplied by 2/'h, gives the rate at which it is scattered out of the initial state. E0 is understood here to be the energy of the state
corresponding to 1f;0. Equation 7 may be simplified by usin~ Eq. 6 for GE(r,r') and by approximating the exact EElr,r') by its lowest-order form Ei(r,r') from Eq.2. Then
Figure 3. A schematic diagram representing a typical STEM configuration. The target is located at impact parameter b with respect to the focused beam and the 1f;*(r) uk(r)uk(r'* )1f; (r') energy analyzer subtends the half-angle 0m at the angle 0 0 x ------WE 1 (r,r') . (8) 0D with respect to the incident beam. E - Ek - E' + i o 0 straightforward way from Eq. 7 in lowest order when one This is a generalization of an expression given in [11] for takes E 1_ Using the Lehmann form, Eq.3, for W, 0
1f;*(r)uk(r)uk(r')'I/J * (r') 3 3 0 0 El= EEJ d rf d r' ------E exp[ik•(r-r')]/13 o k n E k + E n + i 6 (11) 0 0 k [E-Ek + io]
x ( 0 I v( r) I n) ( n I v( r' ) IO) . (9) One finds after evaluating the integral over E' in Eq. 8 by contour integration Excitation of a Target by a Microprobe Consider the excitation of a localized target El= EE f d3r1f; *(r) f d3r''I/J (r') situated near the origin of coordinates by an electron o k n o o prepared in the form of a narrow beam. Figure 3 shows a schematic representation of such a STEM configuration. Represent the electron by the wave packet [7] (olv(r)ln) (nlv(r')\o) exp[ik•(r-r')] X ------,.------(12)
Multiplying Im E~ by 2/v, where v='hk /m, summing 0 (10) over all final states of the electron and singlin& out a particular state In> of the target, one may show [7] that P n q, the resulting probability of exciting the nth state in where tPq is chosen so that at z=0 the packet is transitions of the electron to all possible plane wave final distributed in a narrow probe about the impact states, is parameter b with spatial extension ,..,t:,_about b. The factor in the wave function describing the z-variation is normalized in the large spatial interval of length L. The self-energy of this electron may be evaluated in a
47 R.H. Ritchie, et al.
In obtaining this result one assumes that k >>Q for 0 values of Q for which ¢q is not much smaller than unity, and that recoil corrections to the final energy due to momentum transfers perpendicular to the beam direction may be neglected. This is described in more detail in [7]. (15) Here P c(p) is the probability of exciting the n target by a classical electron with the same velocity at 2 Here R=(O,y,z), =(0,K ,K ), W=E/'h and it is understood impact parameter p and I ¢(p--b) 1 is the probability of y z finding the electron at that impact parameter. Also, that Ew is a time-ordered local dielectric function, such 2 ¢(p) = f d Qexp(iQ• p)¢q/(2ir)2. This quite general that Ew=E_w.1 We also assume that the probe does not result shows that, if all inelastically scattered electrons penetrate into the solid, although it is straightforward to are collected by the energy analyzer, the measured write down a formula for W E(r,r') that accounts for this probability of exciting a given transition may be possibility [13]. computed theoretically as if the beam consisted of a superposition of classical trajectories distributed laterally Neglecting k 2 in the denominator of Eq. 14, to the beam direction according to the probability p 2 summing over all kp and defining a projected, or local, density 1¢(p--b)l . Saying this another way, provided that the spectrometer aperture in the STEM is large self-€nergy, ~6(x), as [11] enough to accept most of the inelastic scattering, classical excitation functions are correct when averaged over a range of impact parameters corresponding to the current distribution in the probe. In [7] partial signal collection is considered in detail. The probability of surface plasmon generation at a planar surface by a probe and where p=(x,y,O),one finds after equating integrands, with collection of scattered electrons by a small spectrometer aperture placed at various angles is evaluated numerically. With off-axis positions improved spatial resolution is obtained. In all detector positions the width of the microprobe emerges from the wave theory as an obvious limit to the obtainable resolution. 2 e2 [ 1-E , ] 1+ Ew exp(-2KX) The Self-Energy of an Aloof Probe +-- I w Exciting Surface Modes x------( 17) ['hvKZ- E' + io] Consider an electron prepared in a state corresponding to the wave packet of Eq. 10 and traveling 2 parallel with and at distance x from a condensed matter The neglect here of a term k /2 m the denominator surface. Assuming that k >>Qin Eq. 10 and using the z 0 should be quite accurate for the high-€nergy electrons approximation of Eq. 11, one finds used in STEM. The imaginary part of Eq. 17 may be expressed simply b1, using Dirac's formula for the denominator, i.e., 1/(w+ia)=P/w--iiro(w), where o(w) is the Dirac delta function, one finds
(18)
exp(-i zk ) exp(ik• [r-r']) exp(iz'k ) 0 0 x----,.------2 2 2 where K is the modified Bessel function of the second L4[~ (k -k -k ) - E' + io] 0 ..:m o p z kind. If we use the electron gas model, 2 (1-t)/(l+t)=w/f[w -(ws-iai] where 'hwsis the surface plasmon energy and a is a positive infinitesimal. Then
1The time-ordered dielectric function has zeros in the where k = (kp, kz) = (kx, ky' kz). W E(r,r' ), the second and fourth quadrants of the complex w-plane such that E = E The causal Ec , familiar from classical propagator for disturbances in the medium, may be w -w w evaluated in various approximations. For a metal, one dielectric theory, has zeros lying in the third and fourth might use the specular reflection model [18], or a hydrodynamical model [4,19]. For simplicity we use a quadrants, such that E~ = (E.'.:_J*. It is straightforward local dielectric model, in which case to construct the time-ordered E from the causal Ec , given the latter either from experiment or in analytical form.
48 Plasmons in STEM Electron Spectra evaluating the integral of Eq. 17 over E' by contour when the trajectory lies outside of the dielectric and in a integration methods, one finds medium with unit dielectric constant. This result is identical with that found in the wave treatment above in Eq. 18 after multiplying Eq. 18 by 2 to go from a probability amplitude to a probability, dividing by 71,to obtain a damping rate and then further dividing by v to get a probability per unit path length. When x<0 and the electron travels in the dielectric, which goes asymptotically, as X--i(I), to the form 2 Rel.: 1 = --e /4x, the classical image potential for an 0 electron at distance x from a perfect conductor [11,20]. One obtains also [21]
(20) (22)
In the approximation leading to Eq. 17, l.:(x) does not The term containing the factor Im(-1 / f ) describes losses depend at all on the form of the microprobe. In general, to bulk modes. The logarithmic term yields the ordinary and in a more detailed treatment [11,22], one finds such a loss rate to volume excitations, qc is a cutoff wave dependence. number, and the K function describes the boundary (or Higher-order terms in the self-energy function 0 have been worked out for slow electrons near a metal "begrenzung") correction to these losses [4]. surface [12]. Here fw is either the time-ordered or causal The form of WE (r,r') for a spherical target dielectric function of the target, and w = 6.E/h, where characterized by a local dielectric function has been given 6.E is the energy loss. In case the region x>0 is filled , in [13] and employed to evaluate Im l.: and the closely instead, with material having dielectric function fo, one related quantity P , the differential probability for losing w w needs only replace the factor Im[(l-f)/(l+f)] by energy hw to the target. Im[(f 0 -f)/(f 0 +f)] to account for the altered surface Classical Dielectric Theory of Excitation excitation function and employ a factor like the bulk response function of Eq. 22 in the region x>0 but with Interest in the use of the low-loss, valence Im(-1/f) replaced by Im(-1/l). excitation region of the spectrum has been recently Equation 21 may be modified [24] to deal with the revived after having been dormant for several years. For case of a beam reflecting from a surface at a small example, one can study surface excitation generated when glancing angle by putting dz=dx/ 0 and integrating over a beam is outside the sample as in reflection electron x. This gives the total probability of surface excitation microscopy (REM) ima&ing or in the SO-{;alled aloof in such a trajectory, counting both incoming and beam [23] studies [24-26] of small particles. Localized, outgoing segments, as low-loss electron spectroscopy in inhomogeneous samples is reemerging as a potentially important new adjunct of electron microscopy. (23) In view of the demonstration above and in Ref. [7] that a classical treatment of energy losses is valid when the energy loss spectrometer collects most of the Comparing experimental data on loss spectra using Cu scattered angular distribution, it is of some interest to surfaces with the function Im[(l-f)/(l+f)]/w, Howie and review work on the local dielectric treatment of the Milne [27] found reasonable agreement between them interaction of electrons, assumed to move on classical despite the presence of an oxide layer. trajectories, with various targets. Generalization of Eqs. 21-23 to take account of A Classical Electron Moving Parallel with a Plane relativistic effects has been made in [28]. However, the Bounded Dielectric relativistic corrections are expected to be small unless One may compute the retarding field, and hence Re( f) becomes large enough that the criterion for the the rate of energy loss experienced by a fast electron 2 2 traveling parallel with, and at distance from, a surface emission of Cherenkov photons ( fv / c > 1) is satisfied using ordinary local dielectric theory. One finds for the for an appreciable range of frequencies. One notes that the function K (2wx/v) diverges probability of exciting the dielectric per unit path 0 length,2 logarithmically as x _, 0 and that Eg_s. 21-22 must lose validity in that limit. Echenique [29] has evaluated the error incurred in using a local f and finds that for x 49 R.H. Ritchie, et al. where · 1 The nonrelativistic theory leadin& to Eq.24 has been extended to oxide-coated spheres [36] as well as to the case where the trajectory passes through the sphere (13]. The results are in qualitative agreement with experiment [36-39], but more work is needed here. Excitation functions for dielectric bodies bounded by more elaborate coordinate systems have been found. These include some allowance for the effect of the support of a spherical particle (38,39] and for interactions between closely-spaced pairs of spherical particles (40,41]. A spheroidal dielectric has been studied (42]. Solutions found for a cylindrical wedge (43,44] have relevance to the case of a fast electron passing near a 10 20 30 ,6 (eV corner of a cube. Results for excitation of a dielectric by a fast Figure 4. Experimental specular beam energy loss electron passing through a cylindrical cavity in the spectrum (49] obtained at the 880 diffraction conditions medium have also been obtained (45--47]. It has been in GaAs with 100 keV electrons. The broken curves are possible to interpret in considerable detail [47] the energy computed for a trajectory A with exponential depth loss spectra obtained experimentally in this geometry. penetration (decay depth = 3.3 nm), for a trajectory D reflecting at the surface and for trajectories traveling Spatial Resolution in Energy Loss Spectroscopy along the surface for 82 nm (E 1) and for 120 nm (E 2) as shown in the inset. An estimate of the effective distance from the track of a swift electron at which excitation of an electronic transition with energy transfer 1i,w will occur if they use experimentally determined Ew values. For can be made on the basis of the duration of the electric x;;; 10 nm the use of relativistic corrections yields better impulse experienced at a given impact parameter by a struck electron. This yields the "cutoff' impact agreement than when the nonrelativistic formula Eq. 21 parameter bc=v/w. For a 15 eV-loss with 100-keV is used. In several papers (27,31-34], workers at the electrons this comes to ,..,7nm. Cavendish Laboratory have analyzed data taken in Cheng (48] has pointed out that this figure is Reflection High Energy Electron Diffraction (RHEED) considerably larger than the spatial resolution of 0.4 nm and Reflection Electron Microscopy (REM). They that has been achieved in some experiments (49] using treated grazing trajectories as a series of short segments the 15-eV loss in Al. He identifies the resolution found parallel to the interface and at different distances from it with the distance traveled by the plasmon before it and have included relativistic and finite aperture effects decays and gets quantitative agreement between his as well as some penetration of the crystal before theory and experiment using reasonable estimates of the reflection. They find that simple dielectric theory seems plasmon group velocity and lifetime. In our view, this to allow fitting of the data in absolute terms. Figure 4 explanation is suspect. It is important to realize that be shows typical reflection energy loss spectra [32] obtained with a GaAs surface as compared with calculations using is an upper estimate of the impact parameter dielectric theory. They have also developed a corresponding to zero scattering angle and thus zero quasi-planar approximation to the excitation of momentum transfer perpendicular to the initial velocity. geometrically complex dielectrics (32]. Larger values of momentum transfer are associated with Excitation of Dielectrics in Other Geometries smaller interaction distances. The strength of the Analytical solutions have been obtained for the excitation at a given energy loss is in general determined excitation produced by a fast classical electron passing by an integral over momentum transfer and ultimately near a spherical dielectric body. When the electron depends on some function of wb/v, such as the function K (2wb/v) in the case of a planar interface (Eq. 20 passes at distance b from the center of a sphere of radius 0 a 50 Plasmons in STEM Electron Spectra available in the excitation spectrum of the plasmon. 3 This where Ti,x;is the momentum transfer perpendicular to the point is discussed below in connection with secondary electron direction. The magnitude of the total electron emission in SEM. momentum Ttk=Tt(x;2+ w2/v2)1/2. The Chang-Raman Transform Secondary Electron Generation Processes and their Degree of Localization In the context of theoretical high energy physics, Chang and Raman [64] have employed a mathematical The secondary electron (SE) signal has long been transformation from momentum to a space-like variable. recognized in scanning electron microscopy as the most It has been advocated for use in radiation physics [63]. useful indicator of surface topography. Recent work in Following their lead one transforms variables from x; to STEM [51,52] has shown that it is possible to obtain SE impact parameter b. This may be done by first signals with 1-nm spatial resolution and 1-eV energy integrating Eq. 26 over w to obtain resolution and that reflection SE images of oxidized Cu show oxide islands and details of their interaction with surface steps. The generation of secondary electrons by d2A-l e2 oodw [ -1 ] I 12 ~=2.2f ~Im -€- = a(x;) fast incident electrons is quite complex, involving d K 1r Ttv o k k, w electron cascade processes created by fast secondaries and the slowing-- The Impact Parameter Representation (IPR) The integrand of this equation is now set equal to The problem of visualizing quantal collisions in the DIMFP in impact parameter space, viz., the space (and perhaps time) variable has been faced by several workers over the years [60-u3]. The lack of a 2 -1 comprehensive theory of an impact parameter d ACR e2 I J 2 . representation for collisions in condensed matter has been ~ = ~ d K exp(1"• b) noted long ago [63]. We approach this problem by using d b 47r Ttv an expression for the probability of interaction of a fast electron with a medium whose response is specified in terms of a dielectric function fk that depends on energy ,w x{foo~lm(!-)}1/212. (27) transfer, Ttw, and the magnitude of the momentum o k k,w transfer, Ttk, to the medium. A more general formulation in terms of the dielectric matrix of the medium is One may easily apply this to analytical forms for fk of possible but we have not yet done this. We write for the ,w differential inverse mean free path (DIMFP) for energy an electron gas. However, for reasons given elsewhere and momentum transfer to the medium by the electron [65], the transformed function described next is preferable to that found using the Chang-Raman method. The Energy-Transfer Transform (26) We have made a more general approach [65] by employing a transform different from, but related to, that of Chang and Raman. We write 3Theoretical determinations of the probability of excitation of a surface plasmon by a STEM electron as a function of distance from a planar dielectric are being carried out by Zabala and Echenique. These account for plasmon damping and dispersion and still show good spatial resolution. 51 R.H. Ritchie, et al. eigenenergies and wave functions of the impurity site electron are hwn and Xn (r) = V where w =w -w , k2=1t2+w2 /v2, and where it is n 0 n o no understood that the integration is to include only the region of (k,w) space corresponding to collective states of the medium. Equation 29 is obtained by using first-order perturbation theory, assuming that the scalar electric potential at the impurity atom may be calculated from -e dielectric theory. In (65] this formulation is considered further. Application of the Impact Parameter Representation In the context of SE we use the Energy Transfer Transform of Eq. 28 above to obtain the distribution in Figure 5. A Feynman diagram representing the process impact parameter of energy deposition in the conduction of creation of a virtual quasi-particle followed by the band of aluminum metal. We take the plasmon-pole excitation of an electron-hole pair in the medium. d.ielectric function of the electron gas to represent the response of the medium (65]. Figure 6 shows the IPR distribution calculated from Eq. 28 for three different Then electron energies. To obtain these results we have integrated Eq. 28 over K by numerical quadrature. The somewhat surprising result is that each of the curves decreases as b increases, going asymptotically as exp(-2wpb/v) when b -1 oo. It turns out that for each of these primary energies the mean value of b, averaged over these distributions is less than 1 nm. For emphasis in plotting the calculated DIMFP values have been 3 2 2 multiplied by 21rhv /e wp . The small fluctuations in these curves correspond to quanta! effects in the plasmon field. These results turn out to be quite insensitive to the damping constant assumed in the dielectric model. Note We term this the "energy transfer transform" since it that propagation and decay of the plasmon are described agrees precisely with the formula obtained by computing in detail in this treatment. It appears that the narrow the energy transferred to the medium at fixed impact spatial resolution of these IPR distributions is due to the parameter, using quanta! dielectric theory, and then presence of relatively large momentum components in the dividing the integrand in the w variable by hw. The interaction spectrum of the swift electron and the inverse mean free path (IMFP) is found by integrating electron gas. over b. The process corresponding to Eq. (28) may be represented by the Feynman diagram of Fig. 5, in which Summary and Conclusions the swift electron creates an excitation in the solid which finally decays through the creation of a real electron-hole A self-energy formulation of the interaction pair. between a STEM electron and a general condensed The Impact Parameter Dependence of Localized Single matter target has been given. It is shown that if most of Particle Transitions Induced by Unlocalized Excitations the inelastically scattered electrons are collected by the A schematic representation of the spatial energy-loss spectrometer, a classical treatment of the dependence of the localization of an initially unlocalized interaction process is valid, even in very inhomogeneous coherent excitation in an extended medium may be situations, provided the results are averaged over a found. Assume that an impurity site in the medium is bundle of trajectories corresponding to the size of the occupied by an electron in an orbital x (r), situated at r. foc~sed probe. (?lassical dielectric theory of target 0 exc1tat10n for vanous geometries is discussed. The Let a swift electron with speed v traverse the medium at problem of spatial resolution in SEM when low-loss impact parameter b relative to the impurity. If the valence excitations occur is addressed by introducing an 52 Plasmons in STEM Electron Spectra 15 :::J- fU I .c +> f1J a.. 12.5 QJ QJ ' ~ C fU QJ 10 E=100 keV X: QJ U'I '- ~ .....C 7.5 ...... fU +> C QJ 5 QJ' ct- ct.... 0 I -0 ~ 'lJ 2.5 ' 0 0 0.25 0.5 0, 75 1 1.25 b-ImPact Parameter-nm Figure 6. A plot of d2A -t/d2b, the inverse mean free path differential in impact parameter b, versus b, for three impact parameter representation of the interaction cross different electron energies, calculated from Eq. 28. For section. It is shown, using a reasonable model of the convenience in plotting, the curves have been scaled by DIMFP, that the experimental SE data showing high multiplying the results from Eq. 28 by the factor spatial resolution may be accounted for by the presence 21rn.v3/e2wp2,where vis the electron speed. of large momentum components in the electron-valence band interaction spectrum. [4] Ritchie RH. (1957). Plasma losses by fast electrons in Acknowledgements thin films. Phys. Rev. 106, 874-881 [5] Powell CJ, Swan JB. (1959). Origin of characteristic Research sponsored jointly by the Office of Health electron energy losses in Al. Phys. Rev. 115, 869-875 and Environmental Research, U.S. Department of Energy [6] Raether H. (1980). Excitation of plasmons and under contract DE-AC05-84OR21400 with Martin interband transitions by electrons. Springer Tracts, Vol. Marietta Energy Systems, Inc., the NATO Collaborative 88, Springer-Verlag, New York. 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Schattschneider: Why is the Energy-Transfer channeling, in The Channeling of Particles, Proceedings transform preferable to the Chang-Raman transform? of a conference at the Institut National des Sciences et Authors: For the reason given in the last answer and Techniques Nucleaires, Gif-sur-Yvette, France, 47-59, because of the fact that the Chang-Raman transform A. Sarazin, Ed. CEA, Service de Documentation, Saclay seems to produce indeterminate results when narrow [60] Bohr N. (1913). On the decrease of velocity of resonances occur in the response function of the medium swiftly moving electrified particles in passing through [61]. matter. Phil. Mag. (6) 25, 10-31 [611 Williams EJ. (1945). Space-time concepts in P. Schattschneider: Is it meaningful to relate some colfision problems. Rev. Mod. Phys. 17, 217-245 momentum transfer, say h/b from a simple reciprocal [62] Neufeld J. (1953). Energy losses of charged particles space argument, to the energy-transfer transform in of intermediate energy. Proc. Phys. Soc. (London) A66, order to access it in diffraction mode EELS? Or else can 489-596 you describe an experimental setup with which the [63] Fano U. (1960). Collective effects in absorption of energy-transfer transform is accessible directly? Can energy from ionizing radiation. Comparative Effects of you possibly do it in a macroscopically homogeneous Radiation. Burton E, Kirby-Smith J, Magee J, Eds. specimen? John Wiley and Sons, New York, 14-21 Authors: The answers to these questions are not clear to [64] Chang NP, Raman K. (1969). Impact parameter us. representation and the coordinate-space description of a scattering amplitude. Phys. Rev. 181, 2048-2055 P. Schattschneider: You state that the oscillations in [65] Ritchie RH, Hamm RN, Turner JE, Wright HA, Fig. 6 are of quanta! nature. In a similar result, given in Ashley JC, Basbas GJ. (1989). Physical aspects of Ref. [61], those oscillations are missing. Can you charged particle track structure. Nuclear Tracks and comment on that? Radiation Measurements, in press Authors: Such oscillations are easy to see in Fig. 12 of [66] Mermin ND. (1970). Lindhard dielectric function in Reference [61] but are more difficult to perceive in Fig. the relaxation-time approximation. Phys. Rev. Bl, 11 because of their small amplitude. 2362-2363 K. Krishnan: The plasmon excitation is, in principle, a Discussion with Reviewers delocalized process. Could you please clarify or explain your definition of an impact parameter for measurements P. Schattschneider: The Feynman diagram Fig. 5 to of this phenomenon? In addition, there is a lot of which you refer after Eq. 28 shows a process where a excitement generated by the development of secondary particular momentum hlc is transferred to an electron detection with polarization analysis. A high electron-hole pair excitation via a plasmon. How does current density source is used and spatial resolutions of this relate to Eq. 28 where a position variable b occurs 10 nm are claimed. Could you comment on the and the momentum is integrated out? applicability of your impact parameters in defining Authors: This diagram illustrates the elemental virtual spatial resolution in the context of measurements of such processes that contribute to the IMFP. To obtain the delocalized emissions? IMFP of Eq. 28 or Eq. 29, it is necessary to transform Authors: See the discussion above. Also, our variables from momentum transfer to impact parameter. interpretation of the results we show in Fig. 6 is that the probability of SE due to the generation of an P. Schattschneider: In the derivation of Eq. 18 you electron-hole pair by a plasmon at impact parameter b assume a geometry as sketched in Fig. 3. The impact from the point at which a fast electron impinges on a parameter b has a precise meaning there. The procedure solid should be narrowly concentrated in a region ;s:1nm leading to the Chang-Raman transform follows closely the derivation of the density autocorrelation function in a from that point for energies of interest in STEM. We feel homogeneous medium given by Van Hove. that these results should be applicable to SE detection Consequently, I would expect Expression 28 to contain with polarization analysis. information on the (radial) distribution of the coherently induced charge at distance b from a randomly selected J. Schou: Does your treatment show that a bulk plasmon point in the homogeneous medium. From that, it seems in aluminum ( and other metals) is very likely to decay to me that you use the work "impact parameter" in two within 0.25 nm from the point of impact of the primary different ways: the "b" in the first part of the paper is a electron? distance between target and probe, whereas "b" in Eq. 28 Authors: As indicated in the discussion of Fig. 6 is a distance between points in the target which oscillate following Eq. 29, for the primary energies studied, decay coherently. What, then, is the meaning of the "impact is expected to occur with a mean value of b -,s:lnm. parameter" b in Eq. 28, and is it different from that leading to Eq. 18? Authors: The derivation leading to the Chang-Raman 55 R.H. Ritchie, et al. J. Schou: What is your opinion of the feasibility of the L. W. Hobbs: In wide-gap insulators, bound previous treatments by Rosler and Brauer and of Chung electron-hole excitons states are spatially highly and Everhart on the plasmon creation and decay in localized. Is there anything you know of which precludes secondary electron emission? In which respects do their decay of spatially-extensive valence plasmons into two treatment differ from yours? (or more) widely-spatially separated single electron-hole Authors: We cannot comment on the details of these bound states? This question is of some interest to those treatments since the calculations are quite intricate. who are concerned with radiolytic damage processes in However, their models of plasmon generation and decay solids in which the efficiency of producing single exciton seem quite reasonable. Of course, we do not attempt to states from other, more probable excitations is especially include the effect of electron transport and cascade in our relevant. work. Rather, our results refer to the spatial distribution Authors: In principle the decay of collective states into of decay of quasiparticles such as plasmons into first two or more electron-hole pairs should occur. However, generation electron-hole pairs. Since the energies of the one suspects that the probability of such processes may electrons generated in these decays are Nl0 eV, the be small compared with the probability of decay into a transport and cascade of such electrons should occur over single electron-hole pair. For example, theoretical [J. C. very small distances ( 56